I can't tell you how many times I asked, "who came up with this deadline?" and I get "someone in the sales team," or something like that.
1. Focus on flying the plane in a manner that uses as little fuel as possible not bothering to even glace at the fuel gauge, watching the air currents and gliding where possible.
2. Turn on autopilot and spend time and effort trying to calculate the fuel you have to see if you make it to shore before going back to flying as conservatively as possible.
Number 2 feels better. Number 1 is more likely to end with a safe landing.
If you have a deadline and won't be getting new resources any time soon, and what purpose does a estimate serve if you are already committed?
An estimate serves the purpose of allowing you to say “well the deadline is in 3 weeks and our best estimate is that this will take 9 weeks. Lets save ourselves the effort, cancel the project, and do something more profitable with our time.
In your analogy, that would be the equivalent of… jumping out of the airplane mid-flight, confident that you can just ride brooms to Iceland because thinking of this as a life-or-death decision is incorrect.
So if it helps you, just imagine here that we're talking about the 99+% case, rather than the extremely rare "real" deadline.
I didn't commit, the sales person did. I'm not in trouble, he and the company is. I can just leave, it is their fault if they make bad deadlines. The only reason they can continue is that engineers don't push back and quit when it happens. You work hard so that the sales person can get his juicy bonus, why should you do that?
With it, sales departments can spend more than they have, because they sell for X and need to pay production, IT and project management Y1, Y2 and Y3. An unrealistic deadline would put them in the negative due to overtime payments, attrition, etc. So would unclear requirements, failing to do their due diligence, selling science fiction...